Complex Number to Rectangular Form Calculator

Convert from polar form r∠θ to rectangular form a + bi. Supports degrees and radians.

Enter Polar Form (r, θ)

Result

Rectangular Form
3 + 4i
a + bi
Real Part (a) 3
Imaginary Part (b) 4
Magnitude (r) 5
Angle (degrees) 53.13°
Angle (radians) 0.9273 rad
Trigonometric Form 5(cos 53.13° + i sin 53.13°)

Step-by-Step Solution

a = r cos(theta), b = r sin(theta)

Converting Polar to Rectangular Form

When a complex number is given in polar form as r∠θ (r at angle θ) or equivalently r(cosθ + i·sinθ), we can convert it to rectangular form a + bi using the fundamental relationships between polar and Cartesian coordinates. The real part a is the horizontal projection and the imaginary part b is the vertical projection of the complex number on the complex plane.

Conversion Formulas

Real Part

The real part is the cosine projection of the magnitude.

a = r * cos(theta)

Imaginary Part

The imaginary part is the sine projection of the magnitude.

b = r * sin(theta)

Rectangular Form

Combine real and imaginary parts into standard form.

z = a + bi = r cos(theta) + r sin(theta) i

Degree to Radian

Convert degrees to radians before using trig functions if needed.

radians = degrees * (pi / 180)

Common Polar-to-Rectangular Conversions

Here are some frequently encountered conversions that are useful to memorize:

  • r∠0° = r + 0i = r (positive real axis)
  • r∠90° = 0 + ri = ri (positive imaginary axis)
  • r∠180° = -r + 0i = -r (negative real axis)
  • r∠270° = 0 - ri = -ri (negative imaginary axis)
  • r∠45° = r(sqrt(2)/2) + r(sqrt(2)/2)i
  • r∠30° = r(sqrt(3)/2) + r(1/2)i
  • r∠60° = r(1/2) + r(sqrt(3)/2)i

When to Use Each Form

Both polar and rectangular forms have their advantages depending on the operation:

  • Addition/Subtraction: Rectangular form is easier -- just add or subtract the real and imaginary parts separately.
  • Multiplication/Division: Polar form is easier -- multiply or divide magnitudes and add or subtract angles.
  • Powers/Roots: Polar form with De Moivre's theorem makes these calculations straightforward.
  • Plotting: Rectangular form directly gives (x, y) coordinates on the complex plane.

Degrees vs. Radians

Angles can be expressed in either degrees or radians. Degrees divide a full circle into 360 equal parts, while radians measure the angle by the arc length on a unit circle. One full revolution is 360° = 2*pi radians. When using a calculator or programming language, always verify which unit your trigonometric functions expect -- most programming languages use radians.

Verification Technique

After converting from polar to rectangular form, you can verify your answer by converting back. If z = a + bi, then:

  • r = sqrt(a² + b²) should match the original magnitude
  • theta = atan2(b, a) should match the original angle (possibly differing by multiples of 360°)

Applications in Engineering

Converting between polar and rectangular forms is fundamental in electrical engineering for analyzing AC circuits. Impedances, voltages, and currents are often expressed in polar form for ease of multiplication, but rectangular form is needed for adding series components. Signal processing, control systems, and electromagnetic theory all rely heavily on these conversions.